We investigate the shift current bulk photovoltaic response of materials close to a band inversion topological phase transition. We find that the bulk photocurrent reverses direction across the band inversion transition, and that its magnitude is enhanced in the vicinity of the phase transition. These results are demonstrated with first principles density functional theory calculations of BiTeI and CsPbI_{3} under hydrostatic pressure, and explained with an analytical model, suggesting that this phenomenon remains robust across disparate material systems.